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Thomas Laffey

Publications and source records attributed to Thomas Laffey.

6 recordsLinked to original sources
arXiv2020

The Karpelevič Region Revisited

We consider the Karpelevič region $Θ_n \subset \mathbb{C}$ consisting of all eigenvalues of all stochastic matrices of order $n$. We provide an alternative characterisation of $Θ_n$ that sharpens the original description given by Karpelevič. In particular, for each $θ\in [0, 2π),$ we identify the point on the boundary of $Θ_n$ with argument $θ.$ We further prove that if $n \in \mathbb{N}$ with $n \ge 2,$ and $t \in Θ_n,$ then $t$ is a subdominant eigenvalue of some stochastic matrix of order $n$.

math.SP↗
arXiv2019

Completely positive factorizations associated with Euclidean distance matrices corresponding to an arithmetic progression

Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties. We exploit those properties to develop completely positive factorizations of translations of those matrices. We show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix. We also discuss completely positive factorizations of such matrices over the integers. Methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties.

math.SP↗
arXiv2015

The effect of assuming the identity as a generator on the length of the matrix algebra

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest number $k$ such that words of length not greater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets $\mathcal S$ and counted as a word of length $0$. In this paper we discuss how the problem changes if this assumption is removed.

math.RA↗
arXiv2007

On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.

math.NT↗